Numerical controller including corner multiple curves inserting unit

ABSTRACT

A numerical controller configured to control a machine tool for machining the workpiece on the basis of a machining program composed of a plurality of blocks includes a corner multiple curves inserting unit. This corner multiple curves inserting unit inserts, between consecutive two blocks, three cubic polynomial curves in which a position, a direction and a curvature are continuous and the distances from these two blocks are within a prescribed allowable tolerance, if a direction or a curvature between these two blocks is discontinuous in the machining program.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a numerical controller capable ofsetting an allowable amount of inward rounding at a corner arisingbetween blocks. More specifically, the present invention relates to anumerical controller having the function to insert, between the twoblocks forming the corner, a plurality of cubic or higher-orderpolynomial curves continuous in position, direction and curvature and nolarger in inward rounding than a preset allowable amount of inwardrounding.

2. Description of the Related Art

In numerical control for controlling a machine tool, a corner is formedin a machining path as the result of a change in the direction ofmovement, when blocks commanding machining are executed continuously.Shock is liable to occur in the machine tool since the velocity of eachaxis, including a moving axis, suddenly changes at the corner.

In general numerical control, acceleration and deceleration is performedin order to suppress such shock. Examples of acceleration anddeceleration methods include pre-interpolation acceleration anddeceleration in which acceleration and deceleration is performed along amachining path before interpolation processing and post-interpolationacceleration and deceleration in which acceleration and deceleration isperformed on each axis after interpolation processing.

In pre-interpolation acceleration and deceleration, a calculation ismade of such a corner velocity as to make the amount of velocity changein each axis no larger than a preset allowable velocity difference, inorder to suppress a sudden change in the velocity of each axis at thecorner. Velocity control is performed in a manner such that a feed rateis decelerated from a position a distance short of the corner in orderthat the feed rate at the corner agrees with the calculated cornervelocity and is then accelerated on and after arrival at the corner.

In post-interpolation acceleration and deceleration, control isperformed so as to locally average the velocity of each axis determinedby pre-interpolation acceleration and deceleration on the basis of time,i.e., so as to suppress a velocity change in each axis, in order tofurther suppress shock arising in the machine tool. As a result,acceleration and deceleration is performed in an overlapping mannerbetween blocks, and therefore, the machining path deviates from acommanded machining path, thus causing an inward rounding error.

The amount of inward rounding varies depending on the difference of arounding angle at the corner, a difference in the moving axis betweenpositions preceding and following the corner, or a time constantrepresenting the characteristics of post-interpolation acceleration anddeceleration. Accordingly, in order to keep the amount of inwardrounding no larger than a given value, the allowable velocity differenceof each axis or the time constant of post-interpolation acceleration anddeceleration needs to be adjusted for each machine or each machiningprogram used.

Japanese Patent Application Laid-Open No. 9-190211 discloses a techniqueto interpolate a curve into a corner in a case where acceleration isdiscontinuous at the corner and make a velocity and an accelerationcontinuous at both ends of the inserted curve. Six unknown quantitiesare required in order to make a position, a velocity and an accelerationcontinuous at both ends of the curve. In this patent document, thisrequirement is satisfied by a single quintic curve.

In addition, Japanese Patent Application Laid-Open No. 10-320026describes an example of inserting a plurality of polynomial curves intoa corner. If the curves to be inserted cubic curves, a direction and acurvature can be made continuous at both ends of the curves by insertingfour cubic curves.

In the technique disclosed in Japanese Patent Application Laid-Open No.9-190211 mentioned above, interpolating higher-order curves into thecorner tends to complicate a path, resulting in difficulty in control.Complicated curves lead to a large curvature at the corner. This patentdocument does not specifically describe the way of determining suchcurves as to make a deviation from an original commanded path to fallwithin an allowable tolerance.

Three cubic curves will suffice for an original direction and curvatureto be continuous. Unfortunately, however, such constraints as to make adeviation fall within the allowable tolerance at the midway point ofcurves to be inserted are added in Japanese Patent Application Laid-OpenNo. 10-320026 mentioned above, as a result, creating four cubic curvesis needed.

SUMMARY OF THE INVENTION

Hence, an object of the present invention is to provide a numericalcontroller for controlling a machine tool, wherein the numericalcontroller is capable of setting the amount of inward rounding at acorner to be no larger than a preset allowable tolerance irrespective ofmachining shapes and commanded velocities, thereby ensuring machiningaccuracy with respect to a commanded shape, making a direction and acurvature at the corner continuous, relieving shock on the machine toolat the corner, consequently enabling control independent ofpost-interpolation acceleration and deceleration, and minimizingdeceleration at the corner.

The present invention allows a velocity and an acceleration to be madecontinuous and a path to be stabilized using a plurality of lower-ordercurves. In the present invention, curves smaller in error than theallowable tolerance are determined by calculation if a corner is presentbetween consecutive linear blocks or by trial in the case ofcircular-arc blocks and other curved blocks. In addition, in the presentinvention, three cubic curves are enough for a direction and a curvatureto be continuous if curves to be inserted into a corner are cubiccurves. Yet additionally, in the present invention, positions at bothends of a curve are given degrees of freedom to determine such curves asto cause an error in the path to fall within an allowable tolerance.Unknown quantities therefore need not be increased.

The numerical controller according to the present invention isconfigured to control a machine tool for machining a workpiece on thebasis of a machining program composed of a plurality of blocks. Thisnumerical controller includes a corner multiple curves inserting unitfor inserting, between consecutive two blocks, three cubic polynomialcurves in which a position, a direction and a curvature thereof arecontinuous and the distances from the two blocks are within a prescribedallowable tolerance, if a direction or a curvature between the twoblocks is discontinuous in the machining program.

In a case where blocks preceding and following the corner into which thecurves are to be inserted are linear blocks and the linear blocksforming the corner are sufficiently long, both the distance from thevertex of a corner to a starting position of the curves and the distancefrom the vertex of the corner to an ending position of the curves aredefined as the same value d, and both first-derivation vectors at oneand the other ends of the curves are defined as the same value |v|, andthe angle of the corner is defined as θ, and an allowable tolerance atthe corner is defined as l, and then the three cubic polynomial curvesmay be determined by evaluating d and |v| on the basis of therelationship among θ, d and l previously evaluated so as to reduce acurvature and a curvature change in the three cubic polynomial curves.

In a case where blocks preceding and following the corner into which thecurves are to be inserted are linear blocks and the linear blocksforming the corner are not sufficiently long, both the distance from thevertex of a corner to a starting position of the curves and an availablemaximum distance from the vertex of the corner to an ending position ofthe curves are defined as the same value d₁, and both first-derivationvectors at one and the other ends of the curves are defined as the samevalue |v|, and the angle of the corner is defined as θ, and then thethree cubic polynomial curves may be determined by evaluating |v| on thebasis of the relationship between θ and d₁ previously evaluated so as toreduce a curvature and a curvature change in the three cubic polynomialcurves.

The three cubic polynomial curves may allow the distance from the vertexof the corner to the starting position of the curves, the distance fromthe vertex of the corner to the ending position of the curves, andfirst-derivation vectors at both ends of the curves to be evaluated bytrial, in a case where at least either one of the blocks preceding andfollowing the corner into which the curves are to be inserted is not alinear block.

If a path of a tool center point on the workpiece is commanded by amachining program in a five-axis machine tool having three linear axesand two rotational axes for controlling tool directions with respect tothe workpiece, three cubic polynomial curves may be inserted into thepath of the tool center point.

According to the present invention, the amount of inward rounding at acorner can be made no larger than a preset allowable tolerance,irrespective of machining shapes and commanded velocities. As a result,machining accuracy for a commanded shape is ensured. In addition, adirection and a curvature are made continuous at the corner, andtherefore, shock on a machine at the corner is relieved. Consequently,control not reliant on post-interpolation acceleration and decelerationis possible, and deceleration at the corner can be minimized. Thiscontributes to shortening a cycle time and simplifying machineadjustments.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects and features of the present inventionwill become apparent from the following description of embodiments takenin conjunction with the accompanying drawings in which:

FIG. 1 is a schematic view illustrating an example of inserting kpolynomial curves into a corner formed by linear blocks;

FIG. 2 is a schematic view illustrating a case where n=3 and k=3, andthe distance from the vertex of the corner into which the curves areinserted to the starting position of the curves and the distance fromthe vertex of the corner to the ending position of the curves areequally defined as d in FIG. 1;

FIG. 3 is a schematic view illustrating an example of inserting threepolynomial curves between blocks in a case where the blocks forming acorner are circular-arc blocks or curved blocks, other than linearblocks;

FIG. 4 is a schematic view illustrating the way machining with threecubic polynomial curves inserted is performed in a tool-head-rotatingfive-axis machine tool;

FIG. 5 is a schematic view illustrating an example of inserting threecubic curves between linear blocks lying on a two-dimensional plane;

FIG. 6 is a schematic view illustrating an example of inserting curvesin a case where the length of at least one of two linear blocks forminga corner is short in Embodiment 1, and therefore, the curves determinedin Embodiment 1 cannot be inserted;

FIG. 7 is a schematic view illustrating an example of inserting threecubic curves into the junction between a linear block and a circular-arcblock lying on a two-dimensional plane;

FIG. 8 is a schematic view illustrating an example of inserting threecubic curves between two cubic curved blocks lying on a two-dimensionalplane;

FIG. 9 is a schematic view illustrating an example of joining a linearblock and a circular-arc block in a three-dimensional space;

FIG. 10 is a functional block diagram of a numerical controller forillustrating an embodiment of the present invention; and

FIG. 11 is a flowchart for describing a processing flow in a cornermultiple curves inserting unit.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Embodiment 1

Assuming that each curve is represented by f_(i)(t) (0≦t≦1) (i=1, 2, 3,. . . , k), in a case where k curves are inserted into a corner formedby blocks of a machining program, then the conditions for the respectivecurves to join with the directions and curvatures thereof beingcontinuous are given by Equations (1) shown below:

$\begin{matrix}{{{f_{i}(1)} = {f_{i + 1}(0)}},{\left. {\frac{}{t}{f_{i}(t)}} \right|_{t = 1} = \left. {\frac{}{t}{f_{i + 1}(t)}} \right|_{t = 0}},{\left. {\frac{^{2}}{t^{2}}{f_{i}(t)}} \right|_{t = 1} = \left. {\frac{^{2}}{t^{2}}{f_{i + 1}(t)}} \middle| {}_{t = 0}\left( {1 \leq i \leq {k - 1}} \right) \right.}} & (1)\end{matrix}$

Assume that the end point of a preceding block is P₀ and the start pointof a following block is P_(k). Also assume that the two ends ofrespective curves to be inserted between these preceding and followingblocks are P_(i-1) and P_(i) (i=1, 2, 3, . . . , k). That is, the startand end points of the curves to be inserted between the preceding andfollowing blocks are assumed to be P₀ and P_(k), respectively. Inaddition, the positions of these points P₀ and P_(k) are assumed to bep_(s) and p_(e), and the directions of the points are assumed to bev_(s) and v_(e). Yet additionally, the curvatures of these preceding andfollowing blocks are assumed to be a_(s) and a_(e). Then, the conditionsfor the directions and curvatures of the curves to be continuous at thejunctions of these preceding and following blocks and the curves to beinserted between these preceding and following blocks are given byEquations (2) shown below:

$\begin{matrix}{{{{f_{i}(0)} = p_{s}},{\left. {\frac{}{t}{f_{i}(t)}} \right|_{t = 0} = {\left. {{v_{s}.\frac{^{2}}{t^{2}}}{f_{1}(t)}} \right|_{t = 0} = a_{s}}}}{{{f_{k}(1)} = p_{e}},{\left. {\frac{}{t}{f_{k}(t)}} \right|_{t = 1} = {\left. {{v_{e}.\frac{^{2}}{t^{2}}}{f_{k}(t)}} \right|_{t = 1} = a_{e}}}}} & (2)\end{matrix}$

FIG. 1 illustrates an example of inserting k polynomial curves into acorner formed by a linear block the end point of which is P₀ and alinear block the start point of which is P_(k). In FIG. 1, f₁(t) tof_(k)(t) represent the respective curves to be inserted, P₀, P₁, . . .P_(k-1), P_(k) represent the edge points of the respective curves,p_(s), v_(s) and a_(s) represent a position, a direction and acurvature, respectively, at the point P₀ (one linear block-side edgepoint of the first curve f₁(t) to be inserted), and p_(e), v_(e) anda_(e) represent a position, a direction and a curvature, respectively,at the point P_(k) (one linear block-side edge point of the kth curvef_(k)(t) to be inserted).

If curves to be inserted are k in number and nth-order polynomialcurves, the number of constraint conditions is 3(k+1) and the number ofunknown quantities is k(n+1). That is, the total number of constraintconditions is 3(k+1), i.e., 3(k−1) constraint conditions from Equations(1) shown above and representing three constraint conditions and sixconstraint conditions from Equations (2) shown above and representingsix constraint conditions. Thus, as illustrated in FIG. 1, thecoefficients of unknown quantities representing the cubic polynomialcurves f_(l)(t) to f_(k)(t) are k(n+1) in number, i.e., (a_(1,n),a_(1,n-1), . . . , a_(1,1), a_(1,0)), (a_(2,n), a_(2,n-1), . . . ,a_(2,1), a_(2,0)), . . . , and (a_(k,n), a_(k,n-1), . . . , a_(k,1),a_(k,0)).

If the unknown quantities are equal to or larger in number than theconstraint conditions, there exists at least one curve which satisfiesthe above-described conditions. Evaluating n and k at which the numberof constraint conditions equals the number of unknown quantities resultsin two cases, i.e., (1) k=1, n=5, and (2) k=3, n=3. Of the two cases,Case (1) is suggested in Japanese Patent Application Laid-Open No.9-190211. The present invention adopts a case in which the number ofconstraint conditions and the number of unknown quantities are equalizedby setting k=3 when n=3 (i.e., Case (2) mentioned above). Consequently,in the present invention, three (k=3) cubic (n=3) polynomial curves areinserted into a corner. In that case, Equations (1) shown above aregiven by Equations (3) shown below, and Equations (2) shown above aregiven by Equations (4) shown below:

$\begin{matrix}{{{f_{1}(1)} = {f_{2}(0)}},{\left. {\frac{}{t}{f_{1}(t)}} \right|_{t = 1} = \left. {\frac{}{t}{f_{2}(t)}} \right|_{t = 0}},{\left. {\frac{^{2}}{t^{2}}{f_{1}(t)}} \right|_{t = 1} = {\left. {\frac{^{2}}{t^{2}}{f_{2}(t)}} \middle| {}_{t = 0}{f_{2}(1)} \right. = {f_{3}(0)}}},{\left. {\frac{}{t}{f_{2}(t)}} \right|_{t = 1} = \left. {\frac{}{t}{f_{3}(t)}} \right|_{t = 0}},{\left. {\frac{^{2}}{t^{2}}{f_{2}(t)}} \right|_{t = 1} = \left. {\frac{^{2}}{t^{2}}{f_{3}(t)}} \right|_{t = 0}}} & (3) \\{{{{f_{1}(0)} = p_{s}},{\left. {\frac{}{t}{f_{1}(t)}} \right|_{t = 0} = {\left. {{v_{s}.\frac{^{2}}{t^{2}}}{f_{1}(t)}} \right|_{t = 0} = a_{s}}}}{{{f_{3}(1)} = p_{e}},{\left. {\frac{}{t}{f_{3}(t)}} \right|_{t = 1} = {\left. {{v_{e}.\frac{^{2}}{t^{2}}}{f_{3}(t)}} \right|_{t = 1} = a_{e}}}}} & (4)\end{matrix}$

The polynomial curves f₁(t), f₂(t) and f₃(t) in that case are given byEquations (5) shown below:

f ₁(t)=a _(1,3) t ³ +a _(1,2) t ² +a _(1,1) t+ _(1,0)

f ₂ =a _(2,3) t ³ +a _(2,2) t ² +a _(2,1) t+a _(2,0)

f ₃(t)=a _(3,3) t ³ +a _(3,2) t ² +a _(3,1) t+a _(3,0)  (5)

Here, consider that three cubic polynomial curves are inserted into acorner in a case where both of preceding and following blocks formingthe corner into which curves are to be inserted are linear blocks andthe linear blocks forming the corner are sufficiently long.

If both the distance from the vertex of the corner (position at whichthe preceding and following linear blocks intersect with each other) tothe starting position of the curves and the distance from the vertex ofthe corner to the ending position of the curves have the same value d, adeviation e from the commanded path of the curves is made smaller thanan allowable tolerance l by adjusting the value of d.

FIG. 2 illustrates a case where n=3 and k=3 in FIG. 1, and both thedistance from the vertex of the corner into which the curves are to beinserted to the starting position of the curves and the distance fromthe vertex of the corner to the ending position of the curves aredefined as the same value d.

Simulation has proved that the magnitudes of first-derivation vectors(v_(s) and v_(e) in FIG. 1) at both ends of curves to be interpolatedbetween the preceding and following blocks are given by Equation (6)shown below. Here, the magnitudes of both v_(s) and v_(e) are defined asthe same value |v|.

$\begin{matrix}{{v} = {\frac{24}{23}\left( {d - \frac{l}{\cos \frac{\theta}{2}}} \right)}} & (6)\end{matrix}$

Simulation has also proved that the corner angle θ and d/l which is aratio of the distance d from the vertex of the corner to both ends ofthe curves to the allowable tolerance l, in such curves as to make acurvature and a curvature change at the corner as small as possible,while keeping the path of the curves within an allowable tolerance, havethe relationship shown in Table 1 below:

TABLE 1 θ d/l 15° 1.32322 30° 1.64481 45° 1.98257 60° 2.35809 75°2.80097 90° 3.35671 105°  4.10374 120°  5.19551 135°  6.98659 150° 10.5364 165°  21.1354

Consequently, given the corner angle θ and the allowable tolerance l, dis determined from Table 1 shown above. By substituting θ, l and d intoEquation (6) shown above, it is possible to evaluate the magnitudes ofthe first-derivation vector |v| at one and the other ends of the curvesinserted into the corner. In addition, from the distance d from thevertex of the corner to the starting position (and the ending position)of the curves and the commanded path, it is possible to evaluate thepositions p_(s) and p_(e) of the point P₀ (the end point of the linearblock preceding the corner) and the point P_(k) (start point of thelinear block following the corner). Yet additionally, from the |v| andthe commanded path, it is possible to evaluate the directions v_(s) andv_(e) at the points P₀ and P_(k). Still additionally, since the blockspreceding and following the corner are linear blocks (0 in curvature),a_(s)=a_(e)=0. From the foregoing, Equations (4) shown above becomedefinite. Then, by solving Equations (3) and (4) as constraintconditions, it is possible to determine the three curves f₁(t), f₂(t)and f₃(t) to be inserted into the corner.

Embodiment 2

If blocks preceding and following a corner into which curves are to beinserted are linear blocks and if the linear blocks forming the cornerare not sufficiently long, it is not possible to secure the distance dgiven by Table 1 shown above. Simulation has proved that in that case,such a proportional relationship as shown in Table 2 below existsbetween d_(l) and the magnitude |v| of the first-derivation vectorsv_(s) and v_(e) at both ends of the curves to be inserted between thepreceding and following blocks, assuming that the maximum availabledistance is d_(l). Accordingly, given the θ and d_(l), it is possible toevaluate |v|. Subsequent steps are the same as those of Embodiment 1,and therefore, will not be described again here.

TABLE 2 θ |v|/d_(l) 15° 0.248086 30° 0.386692 45° 0.473786 60° 0.53251175° 0.573899 90° 0.603851 105°  0.625786 120°  0.641793 135°  0.653196150°  0.660836 165°  0.665231

Embodiment 3

If at least either one of blocks preceding and following a corner intowhich curves are to be inserted is not a linear block, that is, ifblocks forming the corner are circular-arc blocks or curved blocks,other than linear blocks, the following steps are taken.

(1) Choose arbitrary points as the start point (P₀) and end point(P_(k)=P₃) of curves.(2) Evaluate directions v_(s) and v_(e) having a length (α|P₀−P₃|) αtimes the distance between the start point P₀ and the end point 2 ₃, andalso evaluate curvatures a_(s) and a_(e). Note that these v_(s) andv_(e) are vectors having the directions of v_(s) and v_(e) evaluated inEquations (4) and a length of α|P₀−P₃|. As a matter of convenience,these directions are represented by the same symbols v_(s) and v_(e) asused earlier.(3) Determine three cubic polynomial curves f₁(t), f₂(t) and f₃(t). (thesame method as in Embodiment 1 is used, and therefore, will not bedescribed again here).(4) Change the value of a mentioned above in several ways to repeatsteps (2) and (3) described above, and then evaluate f₁(t), f₂(t) andf₃(t), among the curves thus obtained, which minimize the maximum valueof curvatures.

Embodiment 4

A five-axis machine tool having two linear axes and two rotational axesfor controlling tool directions with respect to a workpiece is availableas an example of machine tools. Examples of this five-axis machine toolinclude a tool-head-rotating five-axis machine tool in which tool headsrevolve on two rotational axes, a table-rotating five-axis machine toolin which a table revolves on two rotational axes, and a mixed typefive-axis machine tool in which a tool head revolves on one rotationalaxis and a table revolves on one rotational axis. If the path of a toolcenter point on a workpiece is commanded by a machining program in afive-axis machine tool, three cubic polynomial curves are inserted, forthe path of the tool center point, so that the deviation e of the curvesfrom the commanded path is no larger than an allowable tolerance l. FIG.4 illustrates the way machining with three cubic polynomial curvesinserted is performed in a tool-head-rotating five-axis machine tool.

Numerical Example 1

FIG. 5 illustrates an example of inserting three cubic curves betweenlinear blocks on a two-dimensional plane.

In this numerical example, assume that the angle θ formed by the twolinear blocks is 120° and an allowable tolerance l at a corner is 1.0mm. Also assume that the two linear blocks are sufficiently long.

From Table 1, the distance d from the vertex O of the corner to thestart point of the curves to be inserted is 5.19551 mm. In addition,assuming that the vertex of the corner is an original point, and threecurves to be inserted are expressed by Equations (7) shown below using Xand Y coordinates on the curves corresponding to a parameter t:

f ₁(t)=(−0.0905t ³+3.33t−5.2,0.269t ³)

f ₂(t)=(−0.0969t ³−0.271t ²−3.06t−1.95,−0.0560t ³+0.806t ²+0.806t+0.269)

f ₃(t)=(0.187t ³−0.562t ²+2.23t+0.743,−0.213t ³+0.638t²+2.25t+1.82)  (7)

where, each curve is defined under the condition of 0≦t≦1.

At this time, the deviation e of the curves from the commanded path is1.0 (mm).

Numerical Example 2

FIG. 6 illustrates an example of inserting curves in a case where thelength of at least one of the two linear blocks forming the corner istoo short to insert the curves determined in Embodiment 1. In thisexample, the angle formed by the two linear blocks is assumed to be 120°and the allowable tolerance l at the corner is assumed to be 1.0 (mm).In addition, the shorter of the two linear blocks is assumed to be 3.0(mm) in block length (d_(l)). The distance from the vertex O of thecorner to the start point of the curves to be inserted is 3.0 (mm). FromTable 2 shown above, the magnitude of a directional vector at both endsof the curves at this time is 3.0×0.641793=1.925379. Assuming that thevertex of the corner is an original point, the three curves to beinserted are expressed by Equations (8) shown below:

f ₁(t)=(−0.0522t ³+1.93t−3,0.155t ³)

f ₂(t)=(−0.0560t ³−0.157t ²+1.77t−1.13,−0.0323t ³+0.465t ²+0.465t+0.155)

f ₃(t)=(0.108t ³−0.325t ²+1.29t+0.429,−0.123t ³+0.368t²+1.30t+1.05)  (8)

where, each curve is defined under the condition of 0≦t≦1.

At this time, the deviation e of the curves from the commanded path is0.577 (mm). The reason for the deviation e being sufficiently smallcompared with the deviation in Numerical Example 1 described above isthat d_(l) (=3.0) is smaller than d (=5.19551) in Embodiment 1.

Numerical Example 3

FIG. 7 illustrates an example of inserting three cubic curves into thejunction between a linear block and a circular-arc block lying on atwo-dimensional plane. In this example, assume that the linear block andthe circular-arc block are joined with the directions thereof beingcontinuous. Also assume that the lengths of the linear block andcircular-arc block are sufficiently long, the radius of the circular-arcblock is 50.0 (mm), and the allowable tolerance 1 at the corner is 1.0(mm).

Determining such cubic curves as to make the maximum value of curvaturessmall from the above-described conditions, while adhering to theallowable tolerance, results in Equations (9) shown below:

f ₁(t)=(−2.99t ³+43.0t−22.0,4.23t ³)

f ₂(t)=(−0.167t ³−8.97t ²+34.0t+18.0,−4.36t ³+12.7t ²+12.7t+4.23)

f ₃(t)=(1.14t ³−9.47t ²+15,5t+42.8,0.126t ³−0.377t ²+25.0t+25.3)  (9)

where, each curve is defined under the condition of 0≦t≦1.

At this time, the distances from the vertex O of the corner to the startand end points of the curves to be inserted are 22.0 mm and 70.7 mm,respectively, and the deviation e of the curves from the commanded pathis 0.961 mm.

Numerical Example 4

FIG. 8 illustrates an example of inserting three cubic curves betweentwo cubic curved blocks lying on a two-dimensional plane. In thisexample, consider that the curves are inserted into a corner formed bytwo cubic curves S₁ and S₂ represented by Expressions (10) shown below:

$\begin{matrix}{\mspace{79mu} {{S_{1}\text{:}\mspace{14mu} \left( {{{10t^{3}} - {30t^{2}} + {70t} - 50},{{5t^{3}} + {5t^{2}} - {5t} - 5}} \right)}\mspace{79mu} {S_{2}\text{:}\mspace{14mu} \left( {{{{- 5}t^{3}} - {20t^{2}} + {5t}},{{{- \frac{20}{3}}t^{3}} + {\frac{10}{3}t^{2}} - {10t}}} \right)}{{where},{{{each}\mspace{14mu} {curve}\mspace{14mu} {is}\mspace{14mu} {defined}\mspace{14mu} {under}\mspace{14mu} {the}\mspace{14mu} {condition}\mspace{14mu} {of}\mspace{14mu} 0} \leq t \leq 1.}}}} & (10)\end{matrix}$

In addition, assume that the allowable tolerance 1 at the corner is 1.0(mm). Determining such cubic curves as to make the maximum value ofcurvatures small from the above-described conditions, while adhering tothe allowable tolerance, results in Equations (11) shown below:

f ₁(t)=(−0.0748t ³−0.00340t ²+1.74t−2.40,−0.402t ³+0.0361t²+0.768t−1.13)

f ₂(t)=(0.504t ³+0.228t ²+1.51t−0.737,0.258t ³−1.171²−0.368t−0.728)

f ₃(t)=(0.0558t ³−1.74t ²−0.456t+0.0435,0.0250t ³−0.398t²−1.94t−2.01)  (11)

where, each curve is defined under the condition of 0≦t≦1.

At this time, the distances from the vertex O of the corner to the startand end points of the curves to be inserted are 2.65 (mm) and 4.80 (mm),respectively, and the deviation e of the curves from the commanded pathis 0.957 (mm).

Numerical Example 5

FIG. 9 illustrates an example of joining a linear block and acircular-arc block in a three-dimensional space. In this example, assumethat the linear block moves in a Z-axis direction, whereas thecircular-arc block moves on an XY plane. Also assume that the lengths ofthe linear block and circular-arc block are sufficiently long, theradius of the circular-arc block is 50.0 (mm), and the allowabletolerance l at the corner is 1.0 (mm).

Determining such cubic curves as to make the maximum value of curvaturessmall from the above-described conditions, while adhering to theallowable tolerance, results in Equations (12) shown below:

$\begin{matrix}{\mspace{79mu} {{{f_{1}(t)} = \begin{pmatrix}{0.195t^{3}} \\{0.000127t^{3}} \\{{{- 0.00204}t^{3}} - {1.29t} + 2.6}\end{pmatrix}}{{f_{2}(t)} = \begin{pmatrix}{{{- 0.0621}t^{3}} + {0.586t^{2}} + {0.586t} + 0.195} \\{{0.0{.139}t^{3}} + {0.000380t^{2}} + {0.000380t} + 0.000127} \\{{0.220t^{3}} - {0.00613t^{2}} - {1.30t} + 1.30}\end{pmatrix}}\mspace{79mu} {{f_{3}(t)} = \begin{pmatrix}{{{- 0.134}t^{3}} + {0.399t^{2}} + {1.570t} + 1.30} \\{{{- 0.00112}t^{3}} + {0.0421t^{2}} + {0.0429t} + 0.0148} \\{{{- 0.218}t^{3}} + {0.653t^{2}} - {0.653t} + 0.218}\end{pmatrix}}{{where},{{{each}\mspace{14mu} {curve}\mspace{14mu} {is}\mspace{14mu} {defined}\mspace{14mu} {under}\mspace{14mu} {the}\mspace{14mu} {condition}\mspace{14mu} {of}\mspace{14mu} 0} \leq t \leq 1.}}}} & (12)\end{matrix}$

At this time, the distances from the vertex O of the corner to the startand end points of the curves to be inserted are 2.60 (mm) and 3.14 (mm),respectively, and the deviation e of the curves from the commanded pathis 0.924 (mm).

[Functional Block Diagram]

FIG. 10 is a functional block diagram of a numerical controller forillustrating an embodiment of the present invention.

A command analyzing unit 1 analyzes a machining program and converts theprogram into an executable format. A pre-interpolationacceleration/deceleration unit 2 controls the tangential velocity of amachining path. An interpolation processing unit 3 performsinterpolation processing to output a movement command to each axis.Post-interpolation acceleration/deceleration units 4X, 4Y and 4Z forrespective axes perform post-interpolation acceleration and decelerationprocessing in response to movement commands, and drive and controlservos 5X, 5Y and 5Z for respective axes based on the processed movementcommands for respective axes. Here, a three-axis machine tool isreferred to as an example. The numerical controller of the presentinvention includes a corner multiple curves inserting unit 6 belongingto the command analyzing unit 1 to interpolate curved blocks, so thatthe direction and curvature of the commanded path of the machiningprogram are continuous.

[Processing Flow at Corner Multiple Curves Inserting Unit]

A processing flow at the corner multiple curves inserting unit executedin the numerical controller illustrated in FIG. 10 will be describedusing FIG. 11. Hereinafter, a description will be given according torespective steps.

[Step SA01] Whether or not the corner in question is formed by twolinear blocks is determined. If the corner is formed by two linearblocks (YES), the process proceeds to Step SA02, but if the corner isnot formed by two linear blocks (NO), the process proceeds to Step SA06.[Step SA02] The angle θ formed between the blocks is evaluated.[Step SA03] Cubic polynomial curves no larger in error than a givenallowable tolerance are inserted. d is evaluated from θ, l, and Table 1,|v| is evaluated from Equation (6), and d, f₁(t), f₂(t) and f₃(t) areevaluated using Equations (3) and (4) as constraint conditions.[Step SA04] Whether or not d is greater than the lengths of the blockspreceding and following the corner is determined. If d is greater thanthe lengths of the blocks (YES), the process proceeds to step SA05, butif d is not greater than the lengths of the blocks (NO), the processingis finished.[Step SA05] Cubic polynomial curves in which the distances from thevertex of the corner to both ends of the curves are limited areinserted. |v| is evaluated from θ, d_(l), and Table 2, and f₁(t), f₂(t)and f₃(t) are evaluated using Equations (3) and (4) as constraintconditions, and then this processing is finished.[Step SA06] The start point (P₀) and end point (P₃) of the curves arechosen.[Step SA07] Such cubic polynomial curves as to make a direction and acurvature continuous are inserted. Specifically,

(1) v_(s) and v_(e) having a length (α|P₀−P₃|) α times the distancebetween P₀ and P₃ are evaluated and further a_(s) and a_(e)corresponding to these v_(s) and v_(e) are evaluated.

(2) three cubic polynomial curves f₁(t), f₂(t) and f₃(t) are evaluated.

(3) the value of α is changed in several ways to repeat steps (2) and(3) described above, and then f₁(t), f₂(t) and f₃(t), among the curvesthus obtained, which minimize the maximum value of curvatures areevaluated.

Then, the process proceeds to step SA08.

[Step SA08] Whether or not the inserted curves are no larger in errorthan the allowable tolerance is determined. If the curves are no largerin error than the allowable tolerance (YES), this processing isfinished, but if the curves are larger in error than the allowabletolerance (NO), the process proceeds to step SA09.[Step SA09] The start point (P₀) and end point (P₃) of the curves arebrought closer to the vertex of the corner, and the process returns tostep SA07 to continue this processing.

1. A numerical controller configured to control a machine tool formachining a workpiece on the basis of a machining program composed of aplurality of blocks, the numerical controller comprising: a cornermultiple curves inserting unit for inserting, between consecutive twoblocks, three cubic polynomial curves in which a position, a directionand a curvature thereof are continuous and the distances from the twoblocks are within a prescribed allowable tolerance, if a direction or acurvature between the two blocks is discontinuous in the machiningprogram.
 2. The numerical controller according to claim 1, wherein, in acase where blocks preceding and following the corner into which thecurves are to be inserted are linear blocks and the linear blocksforming the corner are sufficiently long, both the distance from thevertex of a corner to a starting position of the curves and the distancefrom the vertex of the corner to an ending position of the curves aredefined as the same value d, and both first-derivation vectors at oneand the other ends of the curves are defined as the same value |v|, andthe angle of the corner is defined as θ, and an allowable tolerance atthe corner is defined as l, and then the three cubic polynomial curvesare determined by evaluating d and |v| on the basis of the relationshipamong θ, d and l previously evaluated so as to reduce a curvature and acurvature change in the three cubic polynomial curves.
 3. The numericalcontroller according to claim 1, wherein, in a case where blockspreceding and following the corner into which the curves are to beinserted are linear blocks and the linear blocks forming the corner arenot sufficiently long, both the distance from the vertex of a corner toa starting position of the curves and an available maximum distance fromthe vertex of the corner to an ending position of the curves are definedas the same value d_(l), and both first-derivation vectors at one andthe other ends of the curves are defined as the same value |v|, and theangle of the corner is defined as θ, and then the three cubic polynomialcurves are determined by evaluating Ivy on the basis of the relationshipbetween θ and d_(l) previously evaluated so as to reduce a curvature anda curvature change in the three cubic polynomial curves.
 4. Thenumerical controller according to claim 1, wherein the three cubicpolynomial curves allow the distance from the vertex of the corner tothe starting position of the curves, the distance from the vertex of thecorner to the ending position of the curves, and first-derivationvectors at both ends of the curves to be evaluated by trial, in a casewhere at least either one of the blocks preceding and following thecorner into which the curves are to be inserted is not a linear block.5. The numerical controller according to claim 1, wherein if a path of atool center point on the workpiece is commanded by a machining programin a five-axis machine tool having three linear axes and two rotationalaxes for controlling tool directions with respect to the workpiece,three cubic polynomial curves are inserted into the path of the toolcenter point.
 6. The numerical controller according to claim 2, whereinif a path of a tool center point on the workpiece is commanded by amachining program in a five-axis machine tool having three linear axesand two rotational axes for controlling tool directions with respect tothe workpiece, three cubic polynomial curves are inserted into the pathof the tool center point.
 7. The numerical controller according to claim3, wherein if a path of a tool center point on the workpiece iscommanded by a machining program in a five-axis machine tool havingthree linear axes and two rotational axes for controlling tooldirections with respect to the workpiece, three cubic polynomial curvesare inserted into the path of the tool center point.
 8. The numericalcontroller according to claim 4, wherein if a path of a tool centerpoint on the workpiece is commanded by a machining program in afive-axis machine tool having three linear axes and two rotational axesfor controlling tool directions with respect to the workpiece, threecubic polynomial curves are inserted into the path of the tool centerpoint.